Calculus How To

How to Find Critical Numbers (Points)

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Derivatives > How to find critical numbers

A critical number is a number “c” that either:

  • Makes the derivative equal to zero, or
  • Results in an undefined derivative.

Critical numbers indicate where a change is taking place on a graph. For example:
critical numbers


  • An increasing to decreasing point, or
  • A decreasing to increasing point.

The number “c” also has to be in the domain of the original function (the one you took the derivative of).

How to find critical numbers

Finding critical numbers is relatively east if your algebra skills are strong; Unfortunately, if you have weak algebra skills you might have trouble finding critical numbers. Why? Because each function is different, and algebra skills will help you to spot undefined domain possibilities like division by zero. If your algebra isn’t up to par—now is the time to restudy the old rules.


Example question: Find the critical numbers for the following function: x2x2 – 9

Step 1: Take the derivative of the function. Which rule you use depends upon your function type. For this example, you have a division, so you can use the quotient rule to get:
-18x(x2 – 9)2.

Step 2: Figure out where the derivative equals zero. This is where a little algebra knowledge comes in handy, as each function is going to be different. For this particular function, the derivative equals zero when -18x = 0 (making the numerator zero), so one critical number for x is 0 (because -18(0) = 0). Another set of critical numbers can be found by setting the denominator equal to zero, you’ll find out where the derivative is undefined:

  • (x2 – 9) = 0
  • (x – 3)(x + 3) = 0
  • x = ±3

Step 3: Plug any critical numbers you found in Step 2 into your original function to check that they are in the domain of the original function. For this function, the critical numbers were 0, -3 and 3. Let’s plug in 0 first and see what happens:

f(x) = 0202-9 = 0. Therefore, 0 is a critical number.

For +3 or -3, if you try to put these into the denominator of the original function, you’ll get division by zero, which is undefined.

f(x) = 3232-9 = 9/0. Therefore, 3 is not a critical number.


That means these numbers are not in the domain of the original function and are not critical numbers.

That’s it!
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Stephanie Glen. "How to Find Critical Numbers (Points)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/how-to-find-critical-numbers/
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